Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 669 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 11? (a) 4793 (b) 4776 (c) 4797 (d) 4785
2. 2.What is the remainder when 144 is divided by 10? (a) 6 (b) 3 (c) 4 (d) 5
3. 3.What is the units digit of 2^42? (a) 2 (b) 6 (c) 4 (d) 8
4. 4.What is the units digit of 2^45? (a) 8 (b) 6 (c) 2 (d) 4
5. 5.Which of these numbers is divisible by 6? (a) 1156 (b) 1161 (c) 1152 (d) 1150
6. 6.What is the remainder when 515 is divided by 9? (a) 2 (b) 1 (c) 7 (d) 3
7. 7.What is the units digit of 2^48? (a) 2 (b) 8 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 733 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 3
9. 9.What is the units digit of 8^42? (a) 6 (b) 4 (c) 2 (d) 8
10. 10.What is the units digit of 2^30? (a) 2 (b) 8 (c) 4 (d) 6
11. 11.What is the smallest number you must add to 872 to make it divisible by 12? (a) 8 (b) 5 (c) 3 (d) 4
12. 12.What is the units digit of 8^27? (a) 2 (b) 6 (c) 4 (d) 8
13. 13.What is the units digit of 2^38? (a) 2 (b) 6 (c) 8 (d) 4
14. 14.What is the units digit of 8^45? (a) 4 (b) 2 (c) 8 (d) 6
15. 15.What is the remainder when 145 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 6? (a) 3552 (b) 3551 (c) 3555 (d) 3559
17. 17.What is the smallest number you must add to 147 to make it divisible by 12? (a) 3 (b) 8 (c) 9 (d) 10
18. 18.Which of these numbers is divisible by 11? (a) 5005 (b) 5003 (c) 5017 (d) 5007
19. 19.Which of these numbers is divisible by 4? (a) 1965 (b) 1957 (c) 1960 (d) 1962
20. 20.What is the units digit of 3^11? (a) 1 (b) 9 (c) 3 (d) 7
21. 21.Which of these numbers is divisible by 3? (a) 1471 (b) 1473 (c) 1478 (d) 1474
22. 22.What is the units digit of 9^28? (a) 2 (b) 1 (c) 0 (d) 9
23. 23.What is the units digit of 2^25? (a) 8 (b) 4 (c) 2 (d) 6
24. 24.Which of these numbers is divisible by 4? (a) 841 (b) 839 (c) 840 (d) 847
25. 25.What is the remainder when 262 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
26. 26.What is the smallest number you must add to 653 to make it divisible by 10? (a) 3 (b) 7 (c) 8 (d) 6
27. 27.What is the units digit of 4^32? (a) 4 (b) 5 (c) 6 (d) 7
28. 28.What is the units digit of 3^13? (a) 3 (b) 9 (c) 1 (d) 7
29. 29.What is the units digit of 7^45? (a) 9 (b) 3 (c) 7 (d) 1
30. 30.Which of these numbers is divisible by 3? (a) 446 (b) 449 (c) 452 (d) 447
31. 31.Which of these numbers is divisible by 11? (a) 3625 (b) 3632 (c) 3630 (d) 3651
32. 32.What is the smallest number you must add to 242 to make it divisible by 7? (a) 2 (b) 4 (c) 3 (d) 5
33. 33.Which of these numbers is divisible by 6? (a) 1675 (b) 1685 (c) 1680 (d) 1691
34. 34.Which of these numbers is divisible by 6? (a) 1230 (b) 1238 (c) 1232 (d) 1227
35. 35.Which of these numbers is divisible by 8? (a) 888 (b) 890 (c) 897 (d) 883
36. 36.What is the smallest number you must add to 212 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
37. 37.What is the smallest number you must add to 223 to make it divisible by 11? (a) 8 (b) 3 (c) 9 (d) 7
38. 38.What is the units digit of 3^20? (a) 3 (b) 9 (c) 1 (d) 7
39. 39.What is the smallest number you must add to 688 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
40. 40.What is the units digit of 3^38? (a) 3 (b) 9 (c) 1 (d) 7

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